where \(S_i^{\alpha }\)  (\(i=1,2,3\) ) are the spin-1 operators at the \(i\) -th lattice site, \(J>0\)  is the anti-ferromagnetic exchange coupling, \(\Delta \)  quantifies the easy-axis anisotropy, and \(D\)  is the uniaxial single-ion anisotropy parameter, which can be experimentally adjusted by nuclear electric resonance [1]}. One may set \(J=1\)  to fix the energy scale. The full phase diagram for the spin-1 Heisenberg XXZ chain with a uniaxial single-ion anisotropic term was extensively studied [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, which consists of six different phases, i.e., the Haldane phase, the large-D phase, two XY phases, the ferromagnetic phase, and the Néel phase [5]}.
